Productive ToolboxBernoulli Equation Calculator
Apply the Bernoulli principle to calculate fluid pressure, velocity, and elevation. Solve for any unknown variable with step-by-step explanations and unit conversion.
Bernoulli Equation Calculator
Select the unknown variable, enter all known values, and instantly solve using the Bernoulli principle: P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂
Result
Settings & Actions
Solve For
Pressure
Velocity
Elevation
Fluid Properties
Standard: 9.81 m/s² (Earth)
Moon: 1.62 m/s²
What is the Bernoulli Equation?
The Bernoulli Equation is a fundamental principle in fluid mechanics that describes the conservation of energy in a flowing fluid. It states that the total mechanical energy — the sum of pressure energy, kinetic energy, and potential energy — remains constant along a streamline for an ideal, incompressible fluid.
The equation is expressed as: P + ½ρV² + ρgh = constant, or in its two-point form: P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂. This allows engineers to relate pressure, velocity, and elevation at any two points in a flow system.
Named after Swiss mathematician Daniel Bernoulli who published it in 1738, this equation underpins the design of aircraft wings, venturi meters, carburetors, nozzles, and countless other engineering systems where fluid flow is involved.
How to Use This Calculator
Step-by-Step Guide
- 1Select the variable you want to solve for (P₁, P₂, V₁, V₂, h₁, or h₂)
- 2The selected field will be disabled — it will be calculated automatically
- 3Enter all other known values with appropriate units
- 4Select a fluid preset or enter a custom density
- 5Results update instantly as you type
- 6View the step-by-step solution and energy breakdown
- 7Copy, save, or export the result
Key Features
- ✓Solve for any of 6 variables (P₁, P₂, V₁, V₂, h₁, h₂)
- ✓Real-time calculation as you type
- ✓Multi-unit support — Pa, kPa, bar, psi, m/s, ft/s
- ✓Automatic SI unit conversion
- ✓Step-by-step formula substitution
- ✓Energy terms breakdown table
- ✓Fluid presets: Water, Air, Oil, Gasoline, Seawater
- ✓Calculation history with localStorage
- ✓Export results as TXT file
- ✓Swap inputs between Point 1 and Point 2
Example Calculations
| Scenario | Known Values | Result |
|---|---|---|
| Pipe constriction (water) | P₁=200 kPa, V₁=2 m/s, h₁=0, V₂=5 m/s, h₂=3 m, ρ=1000 | P₂ ≈ 170.9 kPa |
| Venturi meter | P₁=150 kPa, P₂=120 kPa, V₁=1 m/s, h₁=h₂=0, ρ=1000 | V₂ ≈ 7.76 m/s |
| Elevation change | P₁=P₂=101325 Pa, V₁=V₂=3 m/s, h₁=0, ρ=1000 | h₂ = 0 m |
| Nozzle exit velocity | P₁=300 kPa, P₂=101.3 kPa, V₁=0.5 m/s, h₁=h₂=0, ρ=1000 | V₂ ≈ 19.9 m/s |
| Height difference | P₁=200 kPa, P₂=180 kPa, V₁=2 m/s, V₂=2 m/s, h₁=0, ρ=1000 | h₂ ≈ 2.04 m |
Common Fluid Densities
| Fluid | Density (kg/m³) | Temperature | Common Use |
|---|---|---|---|
| Water | 1000 | 20°C | Plumbing, hydraulics |
| Seawater | 1025 | 20°C | Marine engineering |
| Air | 1.225 | 15°C | HVAC, aerodynamics |
| Oil (SAE 30) | 876 | 40°C | Lubrication systems |
| Gasoline | 720 | 20°C | Fuel systems |
| Mercury | 13,546 | 20°C | Manometers |
Real-World Applications
Aerodynamics
Aircraft wing lift is generated by pressure differences explained by Bernoulli's principle.
Pipe Flow Systems
Engineers use Bernoulli to size pipes, predict pressure changes, and design water supply networks.
Venturi Meters
Flow measurement devices use the pressure-velocity relationship to calculate volumetric flow rate.
Nozzle Design
Nozzle exit velocities and pressures are calculated directly from the Bernoulli equation.
HVAC Systems
Duct pressure analysis and fan sizing rely on Bernoulli to ensure proper airflow distribution.
Carburetors
Fuel-air mixing in carburetors uses the Venturi effect, a direct application of Bernoulli's principle.
Frequently Asked Questions
What is the Bernoulli equation formula?
P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂, where P is pressure (Pa), ρ is fluid density (kg/m³), V is velocity (m/s), g is gravity (m/s²), and h is elevation (m).
What are the assumptions of the Bernoulli equation?
The equation assumes steady, incompressible, inviscid (frictionless) flow along a single streamline. It does not account for viscous losses, turbulence, or compressibility effects.
Why does pressure decrease when velocity increases?
Because total energy is conserved. When a fluid speeds up (higher kinetic energy), its pressure energy must decrease to maintain the constant total. This is the core of the Bernoulli principle.
Can I use this for compressible fluids like air at high speeds?
The standard Bernoulli equation applies to incompressible flow (Mach < 0.3). For high-speed air or gas flows, a compressible form of the energy equation should be used instead.
What units does this calculator use?
All inputs are converted to SI units (Pa, m/s, m, kg/m³) for calculation. Results are then converted back to your selected output unit. You can mix units freely.
Related Tools
Reynolds Number Calculator
Calculate Reynolds Number instantly to determine fluid flow regime — laminar, transitional, or turbulent. Supports metric and imperial units with real-time results.
Pressure Drop Calculator
Calculate pressure loss in pipes using the Darcy–Weisbach equation. Supports water, air, oil, and custom fluids with metric and imperial units.
Flow Rate Calculator
Calculate volumetric and mass flow rate instantly using engineering formulas. Supports pipe flow, fluid velocity, unit conversion, and real-time calculations online.
Hydraulic Pressure Calculator
Calculate hydraulic pressure, force, piston area, and diameter using Pascal's Law (P = F / A). Supports Pa, kPa, MPa, bar, PSI, Newtons, lbf, and more with real-time unit conversion.